IB Mathematics Applications & Interpretation SL 2.2—functions, Domains, Range, Graphs Notes

Why Functions Exist: A Rule With One Reliable Answer

1. Almost everything you want to predict depends on something else: the cost of a taxi depends on the distance, the height of a ball depends on the time since you threw it.
2. To make a prediction you need a rule that turns each input into a single, dependable output, with no ambiguity about which answer you get.
3. A function is exactly that rule: a machine that takes one input and returns one output, the same output every time you feed in the same input.
4. Applications and Interpretation runs on functions used as models, so this reliable-rule idea is the object you fit to data, graph on your GDC, and read meaning from all course long.
5. Once you trust that a function has one output per input, you can also ask the reverse question, given the output, what input produced it, and that reverse rule is the inverse function.

A Function Sends Each Input To Exactly One Output

1. A function is a rule that assigns to each input value exactly one output value.
2. We write the input as $x$ and the output as $f(x)$, read "$f$ of $x$", so the function $f$ turns $x$ into $f(x)$.
3. The defining condition is the word exactly: one input is never allowed to produce two different outputs.
4. So $y = \pm\sqrt{x}$ is not a function, because a single input like $x=4$ would give two outputs, $2$ and $-2$.
5. A quick graphical check is the vertical line test: if any vertical line meets the graph more than once, that input has more than one output, so the graph is not a function.

Function Notation Names The Rule And Its Input

1. The name of the rule does not have to be $f$, and the input does not have to be $x$; the letters are chosen to fit the context.
   1. $v(t)$ usually means velocity written as a function of time $t$.
   2. $C(n)$ usually means cost written as a function of the number of items $n$.
2. The bracket does not mean multiplication: $C(n)$ is "the cost rule applied to $n$", not $C$ times $n$.
3. To evaluate a function at a number, substitute that number for the input everywhere it appears.

A Function Can Be A Mathematical Model

1. A model uses a function to link real-world quantities so you can describe a situation and predict outcomes.
2. When a function is a model, its domain and range must make sense in context, for example a time variable cannot be negative.
3. On your GDC you enter the model, graph it, and read off values, which is how most AI questions expect you to work with a function.

Domain And Range: The Inputs Allowed And The Outputs Reached

1. The domain of a function is the set of all input values for which the rule gives a real output.
2. The range of a function is the set of all output values the rule actually produces.
3. When no domain is stated, take the largest set of real numbers for which the function is defined.
4. Some rules restrict the domain automatically:
   1. for reciprocal and rational functions, exclude any input that makes a denominator zero;
   2. for square-root functions, the expression under the root must be greater than or equal to zero;
   3. for logarithmic functions, the argument must be strictly positive.
5. The range is usually easiest to read off a graph, so graph the function on your GDC and look at the heights the curve actually reaches.

An Inverse Function Undoes What The Function Did

1. The inverse function reverses the effect of a function: if $f$ sends an input to an output, the inverse sends that output back to the original input.
2. If $f$ takes $a$ to $b$, then its inverse takes $b$ back to $a$.
3. The inverse is written $f^{-1}$, read "$f$ inverse", so the reverse rule applied to $x$ is $f^{-1}(x)$.

Finding The Value f⁻¹(10) By Solving f(x) = 10

1. At SL you treat the inverse informally, and you do not need a formula for the whole inverse to find a single value of it.
2. Finding $f^{-1}(10)$ means finding the input whose output is $10$, which is exactly what solving $f(x) = 10$ does.
3. So solving $f(x) = 10$ and evaluating $f^{-1}(10)$ are two names for the same task.
4. Producing a formula for the whole inverse, $f^{-1}(x)$, using an algebraic method is a Higher Level topic (AHL 2.7).

The Inverse Is The Graph Reflected In The Line y = x

1. Reversing a function swaps the roles of input and output, and on a graph the input and output are the two coordinates of a point.
2. The line $y = x$ is the set of points whose two coordinates are equal.
3. Reflecting any point in that line simply swaps its $x$-coordinate and $y$-coordinate: $$ (a,\,b) \;\longrightarrow\; (b,\,a) $$
4. Here is why that reflection gives the inverse. A point $(a,b)$ lies on $f$ exactly when $f(a) = b$.
5. Because $f^{-1}$ undoes $f$, it sends $b$ back to $a$, so $f^{-1}(b) = a$, which means $(b,a)$ lies on $f^{-1}$.
6. So every point of $f$, reflected in $y = x$, lands on $f^{-1}$: the two graphs are mirror images in that line.
7. This is a special case of the reflections of a graph, the full set of which are a Higher Level topic (AHL 2.8); here the mirror line is $y = x$.

Domain And Range Swap Places

1. Because reflecting in $y = x$ turns every input into an output and every output into an input, the domain and range trade places.
   1. The domain of $f^{-1}$ equals the range of $f$.
   2. The range of $f^{-1}$ equals the domain of $f$.
2. The table below summarises how a function and its inverse line up.

Why Only Reversible Functions Have Inverses

1. For the reflection to be a genuine function, the reflected graph must itself pass the vertical line test.
2. That happens only when the original graph passes the horizontal line test: no horizontal line meets it more than once.
3. A function whose graph passes the horizontal line test is called one-to-one, meaning each output comes from exactly one input.
4. If two inputs share an output, like $x^2$ giving $4$ from both $-2$ and $2$, the reflection would give two outputs for one input, which is not a function.
5. Restricting the domain to force a function to be one-to-one is developed at Higher Level (AHL 2.7).